On inclusion relations for absolute summability

On Inclusion Relations for Absolute Summability

We obtain necessary and (different) sufficient conditions for a series summable | ¯ N, p n | k , 1 < k ≤ s < ∞, to imply that the series is summable |T | s , where (¯ N, p n) is a weighted mean matrix and T is a lower triangular matrix. As corollaries of this result, we obtain several inclusion theorems. Let a n be a given series with partial sums s n , (C, α) the Césaro matrix of order α. If σ...

Necessary and sufficient conditions for inclusion relations for absolute summability

In this paper we obtain appropriate necessary and sufficient conditions for |N, pn|k summability to imply that of |N,qn|s for 1< k≤ s<∞. As in [6] we make use of a result of Bennett [1], who has obtained necessary and sufficient conditions for a factorable matrix to map lk → ls. A factorable matrix A is one in which each entry ank = bnck. Weighted mean matrices are factorable. It will not be po...

Some inclusion theorems for absolute summability

Inclusion of absolute summability domains of matrices Factorable matrices Summability method of Cesàro a b s t r a c t In this work the inclusion relations between absolute summability domains of a normal matrix A and certain factorable matrices are described. Thus, some classes of factorable matrices transforming the absolute summability domain of A into a set of convergent or absolutely conve...

Inclusion Theorems for Absolute Summability Methods

In this paper we have proved two theorems concerning an inclusion between two absolute summability methods by using any absolute summability factor.

On Absolute Cesàro Summability